Problem 105 - College Algebra

Question

The early Greeks believed that the most pleasing of all rectangles were golden rectangles whose ratio of width to height is $$\frac{w}{h}=\frac{2}{\sqrt{5}-1}$$ Rationalize the denominator for this ratio and then use a calculator to approximate the answer correct to the nearest hundredth.

Step-by-Step Solution

Answer

The rationalized form of the ratio is \(2\sqrt{5} + 2\) and its approximate value to the nearest hundredth is 6.47.

Step 1: Rationalize the Denominator

Start by multiplying the ratio by \((\sqrt{5} + 1) / (\sqrt{5} + 1)\) in order to rationalize the denominator. The result of this operation is \(2(\sqrt{5} + 1)\).

Step 2: Distribute the 2

Distribute the 2 in the numerator to get \(2\sqrt{5} + 2\).

Step 3: Use a Calculator

Now, to approximate the answer to the nearest hundredth, use a calculator to evaluate \(2\sqrt{5} + 2\).

Key Concepts

Rationalizing the Denominator

When we encounter a fraction with a square root in the denominator, we use a process called rationalizing the denominator to eliminate the square root. This is important because having a non-rational number in the denominator can be problematic for further calculations and for our understanding of the fraction's value. To rationalize the denominator, we multiply the fraction by a form of 1 that will eliminate the square root. In the context of the golden rectangle ratio, where the original denominator is \( \sqrt{5} - 1 \), we multiply by the conjugate \( \sqrt{5} + 1 \) over itself.

Rationalizing serves not only to simplify calculations but also ensures a form that is generally easier to comprehend. This step is crucial in geometry and algebra, especially when exact values are necessary or when we need to express a measurement in simpler terms. Using the conjugate is effective because when we multiply conjugates \( (\sqrt{a} + b)(\sqrt{a} - b) = a - b^2 \) the radicals eliminate each other, leaving us with an entirely rational number.

Understanding Square Roots

Square roots are a fundamental concept in mathematics; they are the inverse operation to squaring a number. To say the square root of a number is to ask: 'Which number, when multiplied by itself, gives the original number?' For example, the square root of 25 is 5 because \( 5 \times 5 = 25 \) . Square roots can be irrational numbers, which means they can't be written as a simple fraction - the square root of 5, \( \sqrt{5} \) , is an example of such an irrational number.

Irrational numbers are decimals that never repeat or terminate, and when they appear in geometric problems, like the one about the golden rectangle, they can pose a challenge in expressing the exactness of the concept. Understanding how to work with square roots, especially in how they factor into geometric ratios or algebraic expressions, is crucial in higher mathematics.

Numerical Approximation

Numerical approximation is the process of finding a number close to the exact mathematical value which is often impractical to use because it may be irrational or exceedingly complex. We use approximation methods when we need a practical value for real-world applications, or simply to ease calculations and understanding. With the golden rectangle ratio, after rationalizing the denominator and arriving at a new expression \( 2(\sqrt{5} + 1) \), we use a calculator to approximate this value.

By approximating to the nearest hundredth, we essentially round off the decimal to two places, offering a value that's much simpler to visualize and use. This is particularly useful in fields like engineering, physics, and even finance, where the exact mathematical value is less critical than a practical, understandable number that can be used in calculations and measurements.